Class function

In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group G that is constant on the conjugacy classes of G. In other words, it is invariant under the conjugation map on G. Such functions play a basic role in representation theory.

The character of a linear representation of G over a field K is always a class function with values in K. The class functions form the center of the group ring K[G]. Here a class function f is identified with the element ∑ g ∈ G f ( g ) g {\displaystyle \sum _{g\in G}f(g)g} .

The set of class functions of a group G with values in a field K form a K-vector space. If G is finite and the characteristic of the field does not divide the order of G, then there is an inner product defined on this space defined by ⟨ ϕ , ψ ⟩ = 1 | G | ∑ g ∈ G ϕ ( g ) ψ ( g ) ¯ , {\displaystyle \langle \phi ,\psi \rangle ={\frac {1}{|G|}}\sum _{g\in G}\phi (g){\overline {\psi (g)}},} where |G| denotes the order of G and the overbar denotes conjugation in the field K. The set of irreducible characters of G forms an orthogonal basis. Further, if K is a splitting field for G—for instance, if K is algebraically closed, then the irreducible characters form an orthonormal basis.

When G is a compact group and K = C is the field of complex numbers, the Haar measure can be applied to replace the finite sum above with an integral: ⟨ ϕ , ψ ⟩ = ∫ G ϕ ( t ) ψ ( t ) ¯ d t . {\displaystyle \langle \phi ,\psi \rangle =\int _{G}\phi (t){\overline {\psi (t)}}\,dt.} In this setting, the irreducible characters form a Hilbert basis of the Hilbert space of square-integrable class functions, by the Peter–Weyl theorem.

When K is the real numbers or the complex numbers, the inner product is a non-degenerate Hermitian bilinear form.

• Brauer's theorem on induced characters

• Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer-Verlag, Berlin, 1977.